Critical exponents in two dimensions and pseudo-εexpansion

Critical exponents and the pseudo-ǫ expansion

We present the pseudo-ǫ expansions (τ -series) for the critical exponents of a λφ4 threedimensional O(n)-symmetric model obtained on the basis of six-loop renormalization-group expansions. Concrete numerical results are presented for physically interesting cases n = 1, n = 2, n = 3 and n = 0, as well as for 4 ≤ n ≤ 32 in order to clarify the general properties of the obtained series. The pseudo...

Critical Exponents and Dimensions for Elliptic Equations

Polyharmonic Dirichlet Problems: Positivity, Critical Exponents and Critical Dimensions

In this summary we want to present the main results which are contained in the author’s “Habilitationsschrift” (dissertation). It has been written in German and has been submitted to the Faculty of Mathematics and Physics of the University of Bayreuth. Moreover we want to comment on these results, sketch some of the ideas how to prove them and give some background information. For the reader’s ...

Hysteresis Loop Critical Exponents in 6 − ǫ Dimensions

The hysteresis loop in the zero–temperature random–field Ising model exhibits a critical point as the width of the disorder increases. Above six dimensions, the critical exponents of this transition, where the “infinite avalanche” first disappears, are described by mean–field theory. We expand the critical exponents about mean–field theory, in 6 − ǫ dimensions, to first order in ǫ. Despite ǫ = ...

Critical exponents for two-dimensional percolation

We show how to combine Kesten’s scaling relations, the determination of critical exponents associated to the stochastic Loewner evolution process by Lawler, Schramm, and Werner, and Smirnov’s proof of Cardy’s formula, in order to determine the existence and value of critical exponents associated to percolation on the triangular lattice.